FlyinBC,
The older issues should not be a problem at this point, as long as the last 3 to 5 years are clear. If the bycicle incident is recorded in some data base, and that data base happend to be accessed by the firm doing the background check, that could be interpreted negatively even with the dismissal. That might not be fair, but it would be looked at in context of the previous incidents, (if they still show in the data bases) which might then be construed as a potential problem with alcohol. It's very difficult to say exactly what will turn up. If you were not hired, you'd never know exactly why....as applicants are routinely turned down for a wide variety of reasons. In other words, you couldn't assume it was something in the background check because it could have been something else.
If you've mentioned it in your posts, I'm sorry I missed it, but where are you in terms of your training, certification, and experience? If you already have your ratings and say, 1500 hours, it's conceivable that a commuter or regional would consider you.
Your goal in the application process is to get the interview. You have nothing to lose by applying, and being scrupulously honest in disclosing whatever the application requires. You need not disclose something that the application does not ask for at this time. If you are fortunate enough to win the interview, it means that they are interested in you, and would like to meet with you in person to determine if you'd be a good fit in their operation. If, during the interview, there are any questions concerning something disclosed on the application, answer in a straight forward manner while remembering that they're trying to determine whether the background suggests a possiblity of future issues or liabilities. In addition, they must ensure that your background will not prohibit you from obtaining clearance to enter the Air Operations Area under the applicable TSA requirements. You can access these online. Without being defensive, do your best to reassure them that you've learned from your mistakes, and hopefully you're in a position to point out that you have not repeated them for an extended period of time.
Once hired, the background check will begin in earnest. Your offer of employment will specifically state that it is contingent on passing the background check, even though the starting date of employment may preceed it's completion. If you begin employment, and something does come up that has not been previously disclosed, you may not have a chance to discuss the matter to your satisfaction, as your employment status is considered probationary and "at will." This can be particularly problematic for those who have left other employment to pursue their first flying job, in that they can find themselves unemployed with little recourse.
If you are obtaining training and ratings in hopes of professional employment, be advised that crews are subject to random and mandatory drug and alcohol tests, and that there are specific procedures required of a pilot in the event of an arrest for DUI, both on an internal corporate level and with respect to FAA medical certificates.
I hope this answers your questions and has been helpful....best wishes!